Full discretisations for nonlinear evolutionary inequalities based on stiffly accurate Runge-Kutta and \(hp\)-finite element methods
In this very important work, the authors study the convergence of full discretizations by implicit Runge-Kutta and nonconforming Galerkin methods applied to nonlinear evolutionary inequalities. This work is the first contribution, where a class of implicit Runge-Kutta methods is investigated for nonlinear evolutionary inequalities. The authors justify in more detail the employed novel formulation of the fully discrete variational inequality and deduce the needed auxiliary results for stiffly accurate Runge-Kutta and \(hp\)-finite element discretizations. They introduce a simple though characteristic model problem, which the authors serve as an illustration of the general framework of nonlinear evolutionary inequalities. The authors first state the nonlinear parabolic initial boundary value problem involving the \(r\)-Laplacian and free boundary condition of Signorini type and then indicate its reformulation as evolutionary variational inequality. In order to deduce an appropriate time-discrete variational inequality, the authors initiate the procedure for the time-continous problem. The approximations based on the \(hp\)-finite element method are introduced. The authors introduce a general framework of nonlinear evolutionary variational inequalities involving monotone-convex functionals. Moreover, they deduce an existence and uniqueness result as well as a priori bounds for the discrete solution, which are an essential ingredient in the convergence analysis.
Signorini-type initial boundary value problem
scientific article; zbMATH DE number 6387174
scientific article